cbvopab1

Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004) (Revised by Mario Carneiro, 14-Oct-2016) Add disjoint variable condition to avoid ax-13 . See cbvopab1g for a less restrictive version requiring more axioms. (Revised by GG, 17-Jan-2024)

	Ref	Expression
Hypotheses	cbvopab1.1	⊢ Ⅎ 𝑧 𝜑
	cbvopab1.2	⊢ Ⅎ 𝑥 𝜓
	cbvopab1.3	⊢ ( 𝑥 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvopab1	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { ⟨ 𝑧 , 𝑦 ⟩ ∣ 𝜓 }

Proof

Step	Hyp	Ref	Expression
1		cbvopab1.1	⊢ Ⅎ 𝑧 𝜑
2		cbvopab1.2	⊢ Ⅎ 𝑥 𝜓
3		cbvopab1.3	⊢ ( 𝑥 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
4		nfv	⊢ Ⅎ 𝑣 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 )
5		nfv	⊢ Ⅎ 𝑥 𝑤 = ⟨ 𝑣 , 𝑦 ⟩
6		nfs1v	⊢ Ⅎ 𝑥 [ 𝑣 / 𝑥 ] 𝜑
7	5 6	nfan	⊢ Ⅎ 𝑥 ( 𝑤 = ⟨ 𝑣 , 𝑦 ⟩ ∧ [ 𝑣 / 𝑥 ] 𝜑 )
8	7	nfex	⊢ Ⅎ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑣 , 𝑦 ⟩ ∧ [ 𝑣 / 𝑥 ] 𝜑 )
9		opeq1	⊢ ( 𝑥 = 𝑣 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑣 , 𝑦 ⟩ )
10	9	eqeq2d	⊢ ( 𝑥 = 𝑣 → ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ↔ 𝑤 = ⟨ 𝑣 , 𝑦 ⟩ ) )
11		sbequ12	⊢ ( 𝑥 = 𝑣 → ( 𝜑 ↔ [ 𝑣 / 𝑥 ] 𝜑 ) )
12	10 11	anbi12d	⊢ ( 𝑥 = 𝑣 → ( ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ( 𝑤 = ⟨ 𝑣 , 𝑦 ⟩ ∧ [ 𝑣 / 𝑥 ] 𝜑 ) ) )
13	12	exbidv	⊢ ( 𝑥 = 𝑣 → ( ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑦 ( 𝑤 = ⟨ 𝑣 , 𝑦 ⟩ ∧ [ 𝑣 / 𝑥 ] 𝜑 ) ) )
14	4 8 13	cbvexv1	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑣 ∃ 𝑦 ( 𝑤 = ⟨ 𝑣 , 𝑦 ⟩ ∧ [ 𝑣 / 𝑥 ] 𝜑 ) )
15		nfv	⊢ Ⅎ 𝑧 𝑤 = ⟨ 𝑣 , 𝑦 ⟩
16	1	nfsbv	⊢ Ⅎ 𝑧 [ 𝑣 / 𝑥 ] 𝜑
17	15 16	nfan	⊢ Ⅎ 𝑧 ( 𝑤 = ⟨ 𝑣 , 𝑦 ⟩ ∧ [ 𝑣 / 𝑥 ] 𝜑 )
18	17	nfex	⊢ Ⅎ 𝑧 ∃ 𝑦 ( 𝑤 = ⟨ 𝑣 , 𝑦 ⟩ ∧ [ 𝑣 / 𝑥 ] 𝜑 )
19		nfv	⊢ Ⅎ 𝑣 ∃ 𝑦 ( 𝑤 = ⟨ 𝑧 , 𝑦 ⟩ ∧ 𝜓 )
20		opeq1	⊢ ( 𝑣 = 𝑧 → ⟨ 𝑣 , 𝑦 ⟩ = ⟨ 𝑧 , 𝑦 ⟩ )
21	20	eqeq2d	⊢ ( 𝑣 = 𝑧 → ( 𝑤 = ⟨ 𝑣 , 𝑦 ⟩ ↔ 𝑤 = ⟨ 𝑧 , 𝑦 ⟩ ) )
22	2 3	sbhypf	⊢ ( 𝑣 = 𝑧 → ( [ 𝑣 / 𝑥 ] 𝜑 ↔ 𝜓 ) )
23	21 22	anbi12d	⊢ ( 𝑣 = 𝑧 → ( ( 𝑤 = ⟨ 𝑣 , 𝑦 ⟩ ∧ [ 𝑣 / 𝑥 ] 𝜑 ) ↔ ( 𝑤 = ⟨ 𝑧 , 𝑦 ⟩ ∧ 𝜓 ) ) )
24	23	exbidv	⊢ ( 𝑣 = 𝑧 → ( ∃ 𝑦 ( 𝑤 = ⟨ 𝑣 , 𝑦 ⟩ ∧ [ 𝑣 / 𝑥 ] 𝜑 ) ↔ ∃ 𝑦 ( 𝑤 = ⟨ 𝑧 , 𝑦 ⟩ ∧ 𝜓 ) ) )
25	18 19 24	cbvexv1	⊢ ( ∃ 𝑣 ∃ 𝑦 ( 𝑤 = ⟨ 𝑣 , 𝑦 ⟩ ∧ [ 𝑣 / 𝑥 ] 𝜑 ) ↔ ∃ 𝑧 ∃ 𝑦 ( 𝑤 = ⟨ 𝑧 , 𝑦 ⟩ ∧ 𝜓 ) )
26	14 25	bitri	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑧 ∃ 𝑦 ( 𝑤 = ⟨ 𝑧 , 𝑦 ⟩ ∧ 𝜓 ) )
27	26	abbii	⊢ { 𝑤 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) } = { 𝑤 ∣ ∃ 𝑧 ∃ 𝑦 ( 𝑤 = ⟨ 𝑧 , 𝑦 ⟩ ∧ 𝜓 ) }
28		df-opab	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { 𝑤 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) }
29		df-opab	⊢ { ⟨ 𝑧 , 𝑦 ⟩ ∣ 𝜓 } = { 𝑤 ∣ ∃ 𝑧 ∃ 𝑦 ( 𝑤 = ⟨ 𝑧 , 𝑦 ⟩ ∧ 𝜓 ) }
30	27 28 29	3eqtr4i	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { ⟨ 𝑧 , 𝑦 ⟩ ∣ 𝜓 }

Metamath Proof Explorer

Theorem cbvopab1

Proof